Solving Nonlinear Equations By Adaptive Homotopy Continuation

This article introduces and constructively illustrates the concept of an adaptive homotopy for solving systems of nonlinear equations. Standard homotopy methods rely on a passive continuation parameter moving from 0 to 1 along the real line and are stymied if the homotopy Jacobian matrix becomes ill-conditioned along this path. In contrast, an adaptive homotopy replaces the passive continuation parameter by a "smart agent" that adaptively makes its way by trial and error from 0+0i to 1+0i in the complex plane C in accordance with certain specified objectives. The homotopy thus adapts to the physical problem at hand rather than requiring the user to reformulate his physical problem to conform to homotopy requirements. The adaptive homotopy algorithm designed and tested in the current study permits the continuation agent to adaptively traverse a "spider-web" grid in C centered about 1+0i in an attempt to achieve two objectives: (a) short continuation path from 0+0i to 1+0i; and (b) avoidance of regions where the homotopy Jacobian matrix becomes ill-conditioned.Annotated pointers to related work can be accessed at http://www2.econ.iastate.edu/tesfatsi/nasahome.htm